Aside from specific training sets, what distinguishes the capabilities of different AI implementations?
If I understand what you mean correctly then the answer is basically nothing. Fundamentlaly all ML algorithms are ding the same thing, which is to optimize some weights for a certain output (this is true even for non-parametric methods in an implict way, but lets not dive too deep here). The only differences are:

Dataset the model is trained on
Specific dimensionality of inputs and outputs to make them compatible with the input data and the output labels.
Complexity that can be expressed by the model (the number of weights and their structure in layers in the case of Neural Networks).
Changes to the optimization process (gradient clipping, regularization, ecc.)
Structural changes to the architecture that embed assumptions about the specific problem setting (e.g. 2D-Convulutions embed assumptions for images, softmax activations embeds the assumption for classification with probabilities, hidden states embed the assumptions about memory and &quot;fogetting&quot;, ecc...)

Therefore, if you have a &quot;similar&quot; problem setting where you can assume that 3, 4, 5 can be the same without problems, then you can just make the appropriate changes in 2 and change the dataset (1) to get a model that works on something different.
Of course, being able to tell how &quot;similar&quot; a problem is to another and what possible things could be different is quite tricky and requires a lot of knowledge about the domain of Machine Learning and how every algorithm works and why.
In essence, I'm saying that in principle you could take a model and train it for a new setting with very minor and simple changes that don't require a lot of knowledge. However, without the wider knowledge on the field of ML/AI you won't be able to tell if what you are doing is ok and how well can it work in general.